Ab initio characterization of the quantum linear-zigzag transition using density matrix renormalization group calculations

Ions of the same charge inside confining potentials can form crystalline structures which can be controlled by means of the ion density and of the external trap parameters. In particular, a linear chain of trapped ions exhibits a transition to a zigzag equilibrium configuration, which is controlled by the strength of the transverse confinement. Studying this phase transition in the quantum regime is a challenging problem, even when employing numerical methods to simulate microscopically quantum many-body systems. Here we present a compact analytical treatment to map the original long-range problem into a short-range quantum field theory on a lattice. We provide a complete numerical architecture, based on the density matrix renormalization group, to address the effective quantum phi(4) model. This technique is instrumental in giving a complete characterization of the phase diagram, as well as pinpointing the universality class of the criticality.